Formalism in the Philosophy of Mathematics

The guiding idea behind formalism is that mathematics is not a body
of propositions representing an abstract sector of reality but is much
more akin to a game, bringing with it no more commitment to an ontology
of objects or properties than ludo or chess. This idea has some
intuitive plausibility: consider the tyro toiling at multiplication
tables or the student using a standard algorithm for differentiating or
integrating a function. It also corresponds to some aspects of the
practice of advanced mathematicians in some periods—for example, the
treatment of imaginary numbers for some time after Bombelli's
introduction of them, and perhaps the attitude of some contemporary
mathematicians towards the higher flights of set theory. Finally, it is
often the position to which philosophically naïve respondents will
gesture towards, when pestered by questions as to the nature of
mathematics.

The locus classicus of formalism is not a defence of the
position by a convinced advocate, but a demolition job by a great
philosopher, Gottlob Frege. Not that he was attacking a straw man
position: the highly influential diatribe by Frege in volume II of
his Grundgesetze Der Arithmetik (Frege, 1903) is an attack on
the work of two real mathematicians, H. E. Heine and Johannes
Thomae. Moreover, philosophers of mathematics are wont to claim that
the position is still widely adopted by mathematicians. It must
be emphasised, however, that ‘formalism’ in this
sense—the Heine/Thomae position as interpreted by Frege, and its
descendants—is to be distinguished from a more sophisticated
position, Hilbertian formalism. (For more information, consult the
entries on Hibert's program and
the Frege-Hilbert controversy.)

It is the non-Hilbertian approach we will be concerned with in this entry, but
for the sake of completeness, we briefly discuss the Hilbertian
approach. The Hilbertian position differs because it depends
on a distinction within mathematical language between
a finitary sector, whose sentences express contentful
propositions, and an
ideal, or infinitary sector. Where exactly Hilbert drew
the distinction, or where it should be drawn, is a matter of debate.
Crucially, though, Hilbert adopted an instrumentalistic attitude
towards the ideal sector. The formulae of this language are, or are
treated as if they are, uninterpreted, having the syntactic form of
sentences to which we can apply formal rules of transformation and
inference but no semantics. Nonetheless they are, or can be useful, if the ideal sector
conservatively extends the finitary, that is if no proof from finitary
premisses to a finitary conclusion which takes a detour through the
infinitary language yields a conclusion we could not have reached,
albeit perhaps (herein lies the utility) by a longer, more
unwieldy proof. The goal of the Hilbert programme was to provide a
finitary proof of this conservative extension result; most, though not
all, think this goal was proved impossible by Gödel's second
incompleteness theorem.

Returning now to our non-Hilbertian focus, the earlier formalism which
Frege attacked does not divide mathematics into the aforementioned
dual categories of the finitary/contentful, and the
infinitary/essentially meaningless but, on the contrary, treats all of
mathematics in a unitary and heterogeneous fashion. I will use
‘formalism’, then, to refer to the non-Hilbertian formalism
and I will provide an account of the formalist views distilled by Frege from
Heine and Thomae and the criticisms he made of them. These criticisms
are widely believed now to contain conclusive refutations of the
Heine/Thomae approach. But there are a number of post-Fregean views
which seem heavily influenced by, or strongly analogous to, formalism.
I will go through these in turn:

Wittgenstein's views on mathematics, primarily the conception to
be found in his Tractatus Logico-Philosophicus;

formalism as found in the
Logical Positivists, particularly Carnap;

Goodman and Quine's nominalist formalism, and

Haskell Curry's version of formalism.

I will conclude with an overall assessment of the prospects for
formalism in contemporary philosophy of mathematics.

Frege does not extract a unified, consistent position from the work of
Heine and Thomae, and much of his criticism is devoted to showing that
they inconsistently slip into modes of thought which are only
appropriate for ‘contentful’ arithmetic, which Frege
takes to be a body of truths expressed by utterances in which
numerical expressions designate abstract referents independent of the mind
(or at least any particular individual's mind). Heine and Thomae
talk of mathematical domains and structures, of prohibitions on
what may be uttered (e.g. against writing ‘3÷0’ which
is deemed, in some special sense, meaningless), of numbers being
greater than less than one another (rather than physical marks being
larger or smaller, darker or lighter)—all things which
make no sense if arithmetic is a theory of marks and their physical
properties or is just a body of transformations of referent-less
symbols. (Heine however reserves, with Kronecker, a special place for
arithmetic, treating it in non-formalist fashion; this position may
thus be more coherent than Thomae's. See Simons (2009) especially
pp. 293-6.)

Nonetheless two distinct positions emerge from the material Frege
works over, doctrines which Resnik (1980 p.54), and likewise Shapiro
(2000, pp.41–48) describe as term formalism and
game formalism, respectively. The term formalist views the
expressions of mathematics, arithmetic for example, as meaningful, the
singular terms as referring, but as referring to symbols such
as themselves, rather than numbers, construed as entities distinct
from symbols. Thus Heine writes:

I define from the standpoint of the pure formalist and call
certain tangible signs numbers. (Frege, 1903/80 §87, p.
183).

The game formalist sticks with the view that mathematical utterances
have no meaning; or at any rate the terms occurring therein do not pick
out objects and properties and the utterances cannot be used to state
facts. Rather mathematics is a calculus in which ‘empty’
symbol strings are transformed according to fixed rules. Thomae puts it
this way:

For the formalist, arithmetic is a game with signs which are called
empty. That means that they have no other content (in the
calculating game) than they are assigned by their behaviour with
respect to certain rules of combination (rules of the game) (Frege,
1903/80 §95, p. 190).

Thomae also remarks “the formal standpoint rids us of all
metaphysical difficulties” (ibid. p. 184). One main
motivation for formalists, then, is to block, avoid, or sidestep (in
some way) any ontological commitment to a problematic realm of
abstract objects. For standard mathematics entails a plethora of
theorems affirming the existence of infinite realms of
entities- numbers, functions, sets and so forth, entities which do not
seem to be concrete. Formalists, in general, wish to divest themselves
of any commitment to these domains which do, indeed, seem hard to
fit into a thoroughly naturalistic conception of reality.

Frege concentrates most of his fire on the term formalist
pronouncements of his targets; but game formalism is the only
game in town for the anti-platonist worried about the ontological
commitment to a realm of abstract objects. For term formalism treats
mathematics as having a content, as being a kind of syntactic theory;
and standard syntactic theory entails the existence of an infinity of
entities—expression types—which seem every bit as
abstract as numbers. Indeed, as Gödel's arithmetisation of
syntax showed, the elements and inter-relationships of standard formal
syntax can modelled as an infinite substructure inside the standard
model of arithmetic.

Frege mercilessly exposes the inadequacies of Heine and
Thomae's position—their confusions as they slip from
term to game formalism; their conflation of sign and signified; the
fact that they do not set out an account of the syntax and proof theory
which is remotely adequate as an account of the mathematics with which
they deal; their hopeless attempts to extend their position from
arithmetic to treat of analysis and real numbers, by this stage in
mathematical history construed by Weierstrass, Cantor and others not
geometrically but as infinite sequences. Thus Frege writes:

In order to produce it [an infinite series] we would need an
infinitely long blackboard, an infinite supply of chalk, and an
infinite length of time. We may be censured as too cruel for trying
to crush so high a flight of the spirit by such a homely objection;
but this is no answer. (p. 219)

Now Frege, himself, ironically, had revolutionised mathematics by
introducing hitherto unprecedented standards of rigour in the
formalisation of mathematical theories. He recognised (§90, pp.
185–6) that one could improve vastly on Heine and Thomae by treating
mathematical theories, their language, axioms and rules, as formal,
mathematical objects in their own right. This is exactly what the
Hilbert programme set out successfully to achieve, creating the new
disciple of metamathematics.

However Frege lays down very stiff challenges even for a rigorous game
formalist fully equipped with the techniques and results of
metamathematics. Such a theorist gives us a characterisation of a
language, for example by setting out what the basic elements are-
primitive symbols, and strings thereof- and then gives a recursive
specification of which strings count as well-formed. Similarly we will
be given a rigorous specification of which arrangements of well-formed
formulae count as proofs in a given system, and of what theorem they
prove in each case. If strings such as ‘3+1=0’ or
‘3 > 2’ come out as provable in the system (arithmetic
modulo 4, say) then that is enough to count them as correct utterances
of the system. No further issue of truth need arise; nor do we need to
assume that there is only one system for a given set of symbols. Nor,
furthermore, need we assume that each such system is complete (though
Frege took Thomae to task for the incompleteness, massive though
readily rectifiable, of his arithmetic calculus). We need make no
assumption that the numerals in these strings refer to anything
outside the system, indeed we need not assume they refer to anything
at all. (This game formalist, then, is not subject to the objection
that ‘3 > 2’ should come out as false,
on any legitimate formalist reading of ‘>’; no need to
think of the numerals as referring to concrete marks and
‘>’ meaning physically greater in size.)

Such a game formalist is a more worthy opponent for the platonistic
Frege to tackle, but there are two main objections he sets out which
still apply to this more sophisticated position. The first is the
question of applicability: if mathematics is just a calculus
in which we shuffle uninterpreted symbols (or symbols whose
interpretation is a matter of no importance), then why has it been
applied so successfully—and in so many ways, to so many
different things—ordinary physical objects, sub-atomic
objects, fields, properties, and indeed from one part of mathematics to
another (we can count the number of dimensions in a pure geometric
space)? Frege writes:

It is applicability alone which elevates arithmetic from a game to
the rank of a science. (§91 in Frege 1903/1980 p. 187)

Secondly, Frege quite rightly and insistently distinguishes on the
one hand the ‘game’—arithmetic, set theory,
topology or whatever—treated simply as a mathematical
object in its own right, a formal system, and, on the another, the theory
of the game. “Let us remember that the theory of the game must be
distinguished from the game itself” (§107, p. 203). Thus in
the ‘game’ of trigonometry we might derive

sin2θ + cos2θ = 1

from the Pythagorean Theorem. In the metatheory we can prove:

⊢ ⟨sin2θ + cos2θ = 1⟩,

the claim that the formula with such and such a code in the
mathematical representation of the syntax (the code represented in the
meta-meta-theory here by ‘⟨sin2θ +
cos2θ = 1⟩’) is provable. Likewise in the
meta-theory we can prove lots of other things about proof and
refutation, for instance we may be able to show that lots of
sentences are neither provable nor refutable.

The problem this raises for the formalist is this: the metatheory
is itself a substantial piece of mathematics, ostensibly committed to
an infinite realm of objects which are not, on the face of it,
concrete. Tokens of the expressions of the object language game
calculus may be finite—ink marks and the like; but since there
are infinitely many expressions, theorems and proofs, these themselves
must be taken to be abstract types. At best, the formalist can achieve
no more than a reduction in commitment from the transfinite realms of
some mathematical theories, such as set theory, to the countably
infinite, but still presumably abstract, realm of arithmetic, wherein
the syntax and proof theory of standard countable languages such as
those of standard set theory, can, as Gödel showed, be
modelled.

Can formalism be developed in such a way as to surmount these two
crucial objections, the problem of applicability and the problem of
the metatheory, as I will call it? (Not that these are the only
objections to formalism, but they are two fundamental ones.) Because
Frege's critique did not quash all formalistic impulses in later
philosophers of mathematics, we shall look, now, at future
developments, to see how they fare.

Wittgenstein was a keen student of Frege's work, directed to
further his studies under Bertrand Russell by Frege himself in the
course of a visit he made to see Frege in Jena. One might, then, think
him inoculated against formalism. But definite formalistic elements
surface in Wittgenstein's Tractatus.

True, the Tractatus is a notoriously difficult work to
interpret. Even the question as to whether the main portion of the
book, essentially all of it other than the ‘frame’ of the
preface and the end, is to be taken as a serious attempt to present a
metaphysics is controversial. If we leave that hermeneutic controversy
aside, and consider the metaphysics which is offered to us, the
formalistic aspects one finds are twofold. Firstly, mathematical
sentences are said to express ‘pseudo-propositions’, and
so are devoid of truth value (only contingent propositions have a
truth value). Secondly, mathematics is described as a
‘calculus’ which is not to be used to represent the world
as it is in itself but whose value is entirely instrumental. To be
sure, the most explicit statement of this is not in the
Tractatus but in comments Wittgenstein wrote on Ramsey's copy
of it:

The fundamental idea of math. is the idea of calculus
represented here by the idea of operation The beginning of
logic presupposes calculation and so number. Number is
the fundamental idea of calculus and must be introduced as
such (Lewy, 1967, pp. 421–2).

In the Tractatus proper we do get the idea that
mathematical propositions are mere instruments (all of mathematics, not
just an ‘ideal’ fragment, as in Hilbert):

Indeed in real life a mathematical proposition is never what we
want. Rather, we make use of mathematical propositions only in
inferences from propositions that do not belong to mathematics to
others that likewise do not belong to mathematics. (In philosophy
the question, “What do we actually use this word or this
proposition for?” repeatedly leads to valuable
insights.) Tractatus ¶6.211.)

This idea is prefaced by the statements:

Mathematics is a logical method. The propositions of mathematics are
equations, and therefore pseudo-propositions. (ibid, ¶6.2)

A proposition of mathematics does not express a thought. (ibid, ¶6.21)

Care must be taken, however. Wittgenstein distinguishes utterances
which are sinnlos, which lack sense (including logical
tautologies and contradictions here) from those which are
unsinnig, nonsensical and it is not clear into which class
mathematical utterances fall. One might well think that the game
formalist should treat mathematical utterances, on that view just
strings of meaningless marks, as unsinnig, not just
sinnlos. One clear difference from game formalism however is
this: for Wittgenstein mathematics should not be conceived of as a
calculus separate from other uses of language. Rather he attempts to
show that parts of arithmetic, at least, can be seen as grounded in
non-mathematical uses of language. Frege by contrast, whilst arguing
that a proper account of arithmetic (and analysis) should show how its
generality enables one to give a uniform account of multifarious
different applications (cf. Dummett, 1991 Chapter 20) also argued
strongly for the view that mathematical utterances have a meaning
independent of, in conceptually prior to, their use in
applications.

Wittgenstein attempts no theory of mathematics in the
Tractatus beyond arithmetic, a rather narrow fragment of
arithmetic at that. The theory clearly shares the anti-platonism of the
game formalist. There are no numbers, arithmetic is to be construed as
a calculus in which one manipulates exponents or indices of operators.
What is an operator? Wittgenstein distinguishes operator terms from
function terms. A perhaps uncharitable reading of the distinction is
this. Wittgenstein held that two occurrences of a function term
f applied to different strings t and u have
different meaning. That is, ‘f’ in
“f(t)” does not mean the same as the outermost
f in “f(f(t)”; this is supposed
to be the basis of the solution to Russell's paradox (3.333). In
particular ‘the father of’ in “the father of
John”, means something different there compared to its outermost
occurrence in “the father of the father of John”. There can
be no genuine iterated application of functions in other words. But
then, retreating somewhat from this bizarre and unwarranted conclusion,
Wittgenstein wished to exclude certain functions, particularly
functions on propositions or related entities, from the doctrine;
these favoured functions he calls ‘operators’ (5.251).

Sentential operators are conceived as mapping not signs, nor
inscriptions, to other signs and ascriptions but rather propositions,
(in Wittgenstein's rather course-grained sense of the term) to
propositions. On Wittgenstein's account of proposition, repeated
application of an operation such as negation … p,
~p, ~~p … may take one back to an earlier
point. Nonetheless Wittgenstein attempts to explicate arithmetic in
terms of sentential operators applied to non-mathematical language. In
slogan form, numbers are exponents of operations (ibid.,
¶6.021). Thus where Ω is schematic for an operator and
Ωp (or Ω(p)) for its application to a
proposition, then we can view the series

p, Ωp, ΩΩp,
ΩΩΩp, ΩΩΩΩp,
…

as the starting point for a ‘definition’ of numbers, to
be had by rewriting it as

Ω0p, Ω0+1p,
Ω0+1+1p,
Ω0+1+1+1p,
Ω0+1+1+1+1p, ….

Here, then, we have infinitely many schematic rewrite rules.
Indices such as ‘0+1+1+1+1’ can be abbreviated
by numerals in the obvious fashion, with
‘0+1+1+1+1’ being abbreviated
‘4’ and so on.

Wittgenstein's examples show (though he did not explicitly state
this) that addition of two number/exponents Ωnp +
Ωmp (likewise Ωn+mp) is
given by the rule:

Ωnp + Ωmp ⇒
Ωn(Ωmp)

telling us we may replace the expression on the left in a
formula by the one on the right.

This is what underlies the ‘correctness’ of identities
such n + m = r except that for Wittgenstein
no such identity expresses a truth. On his account, the identity sign
disappears in a full analysis of language, wherein sameness and
distinction are shown by sameness and distinctness of names, no two of
which refer, in fully analysed language, to the same object (this view
gives grounds for interpreting mathematical utterances in the
Tractatus as unsinnig). Wittgenstein himself did not
trouble to show that abandonment of an identity sign would not cripple
the expressive power of language but others such as Hintikka (1956)
and Wehmeier (2004) have done so. What is left in the underlying
language, with identity excised, are substitution rules
(Tractatus ¶ 6.23).

These have to be interpreted in a general, and schematic, fashion.
Thus when we plug in ~ for Ω, we find that (Wittgenstein had no intuitionist
scruples here) a double application ~~p takes us back to
(has the same sense as) p. But this does not ground the
truth of 2 = 0, since for many other operations ΩΩp is not
equivalent to p. On the other hand

ΩΩ(ΩΩΩp) always has the same
sense as ΩΩΩ(ΩΩp)

Wittgenstein implicitly assumes suitable rules for the interaction
of brackets with operators, in particular, generalised associativity.
(In fact, he uses a mix of brackets and the notation
Ω′p to express Ω(p).)

Since an equation
Ωnp=Ωmp
is, in its underlying logical form, not a universal generalisation
∀n,m(Ωnp =
Ωmp) but a purely schematic
generalisation, there is no form
∃n,m(Ωnp
≠ Ωmp) with which we can express
inequality, even if we can make sense of ‘≠’. Nor can
we express the inequality n ≠ m schematically as
the holding of the inequivalence of
Ωnp from
Ωmp for every choice for
Ω. Otherwise 2 ≠ 0 would fail since ~~p is
equivalent to p. The Tractarian theory cannot handle
inequalities.

So much for addition and the limitations of its account of that
operation. What of multiplication? Wittgenstein does define it,
at ¶6.241, by:

Ωn×mp ⇒
(Ωn)mp

but to grasp this as a general principle we need to know how to
interpret the notation (Ωn)m.
In more conventional mathematics, one might simply define
(xn)m as
xn×m but
clearly this (or rather the equivalent for the interaction of
exponents of operators) would introduce a circularity into
Wittgenstein's account. Alternatively one could appeal to a recursive
theory of exponents
– am×0= a,
am×(n+1)
= am
+ am×n. Since the
principle of induction needed to show that the recursion is coherent
features nowhere in Wittgenstein's system, these rules would
presumably have to be taken as primitive.

Overall, then, Wittgenstein in the Tractatus gives us no
account of mathematics in general other than for a fragment of
arithmetic, basically positive identities involving only addition. And
there he denies that the sentences express propositions with truth
values. Perhaps his account could have been developed further and more
plausibly, despite the difficulties noticed above (but for some
scepticism on that front, see Landini, 2007). If not, we have the
choice of either abandoning all of mathematics except, at most, a
fragment of the arithmetic of addition; or else of rejecting the
Tractarian account. One does not need to be slavishly
uncritical of contemporary mathematics to see what the reasonable
option is here. (Admittedly, rejecting the Tractatus account
is also the option Wittgenstein himself seems to adopt at the end of
the book; here then we enter the issue of what the point is of taking
us through such a bizarre and unconvincing theory in order to throw it
away at the end.)

Wittgenstein's later work on philosophy of mathematics, such as
the Remarks on the Foundations of Mathematics 1956/1978), for a
long time attracted even less approval than the Tractarian
account, though recently philosophers such as Juliet Floyd and Hilary
Putnam have come to its defence as an interesting and informed account
of mathematics (Floyd/Putnam, 2000). Its themes include the rejection
of the actual infinite (indeed the tendency in his writings is
strongly finitist); the denial that undecidable sentences are
meaningful; a rejection of Cantor's powerset proof; the idea that
proof discovery changes the very meaning of the terms involved; and
other very radical ideas. Among them we find a continued adherence to
formalist motifs:

In mathematics everything is algorithm and nothing
is meaning; (Philosophical Grammar, 468).

Another persistent theme in Wittgenstein's thought is that the meaning of mathematics
resides entirely in its utility in non-mathematical applications. But
there is no systematic theory of how this applicability comes about, no
proof of a conservative extension theorem, for example, showing how
application of mathematical calculi to empirical premisses will
never lead us to derive an empirical conclusion which does not follow
from those premisses. And there is no resolution of the problem of the
metatheory. (On the other hand, we should observe that these notes of
Wittgenstein on the philosophy of mathematics were not published by
him, but by others after his death.)

Wittgenstein greatly influenced the positivists. The
‘official’ positivist theory of mathematics, as it were,
is not a formalist one. Mathematical theorems express truths, albeit
in a special way: true by virtue of meaning alone. The most
influential positivist has been Carnap, if one does not classify Quine
as a positivist (but Quine's views, in the 1930s at any rate, were
very close to Carnap's, indeed arguably Quine remained truer to
Carnap's radical empiricism than Carnap did). And one can certainly
discern strong elements of formalism in some of Carnap's writings, for
instance in Logische Syntax der Sprache (1934/37) and
‘Empiricism, Semantics, and Ontology’ (1950/56).

The former book was translated into English as The Logical
Syntax of Language in 1937. In it, Carnap argued that the correct
method in philosophy is to engage in conceptual analysis conceived of
as ‘logical syntax’, roughly speaking syntax proper and
proof theory. To address philosophical differences, one proposes
regimenting the disputed positions in formal languages or
‘frameworks’ which include a system of axioms and
rules of proof; given these, some sentences are
‘determinate’, are provable or refutable. These are the
analytic, and contradictory, sentences, relative to that framework. How
do we choose which system to adopt? Carnap's Principle of
Tolerance (1934/7 p. 52) allows us to adopt any system we
wish:

In logic, there are no morals. Everyone is at liberty to
build up his own logic, i.e. his own form of language, as he wishes.
[Italics in original]

Carnaps extends this unbridled permissiveness to mathematics:

The tolerant attitude here suggested is, as far as special
mathematical calculi are concerned, the attitude which is tacitly
shared by the majority of mathematicians.

Any such calculus can count as a piece of mathematics, even an
inconsistent one. By downplaying or outright discarding semantic
notions, we simply bypass traditional ontological disputes concerning
the nature of the entities mathematics is ‘about’. The only
issue is the pragmatic utility, or otherwise, of any given mathematical
calculus.

A number of concerns arise here. How can Carnap distinguish between
empirical, scientific theories, and mathematical ones? Secondly, if
pragmatic utility is primarily a matter of empirical applications, how
does the Carnapian formalist know that a given calculus will
conservatively extend empirical theory, how can this be known without
appeal to meaningful mathematical results? Carnap writes:

The formalist view is right in holding that the construction of the
system can be effected purely formally, that is to say without
reference to the meaning of the symbols; … But the task which
is thus outlined is certainly not fulfilled by the construction of a
logico-mathematical calculus alone. For this calculus does not
contain … those sentences which are concerned with
the application of mathematics … For instance, the
sentence “In this room there are now two people present” cannot
be derived from the sentence “Charles and Peter are in this room
now and no one else” with the help of the logico-mathematical
calculus alone, as it is usually constructed by the formalists; but
it can be derived with the help of the logicist system, namely on
the basis of Frege's definition of ‘2’. (Carnap 1934/37,
p. 326)

adding (italics are Carnap's) “A structure of this
kind fulfils, simultaneously, the demands of both formalism and
logicism.”

But what is to stop us freely stipulating, in accordance with the
principle of tolerance, ‘bridge principles’ for operators
which proof-theoretically behave like numerical operators—
‘the number of φ's’—via their definitional
occurrence in formulations of arithmetic axioms and where the bridge
principles include the likes of:

That is, we link the numeral for zero with a sentence stating there
is exactly one entity of the appropriate type, the numeral for one with
a sentence stating there are no such entities. If we do so, add the
rules for standard decimal arithmetic, and then try to apply this
calculus, disaster will ensue; but do we not need a contentful
conservative extension result to show that for the calculi we do use,
no disaster can occur?

Gödel's incompleteness theorems pose very difficult problems for
Carnap in this and other regards. The first incompleteness theorem
tells us that in any (ω-) consistent formal theory whose theorems are
recursively enumerable and which entails a certain (rather limited)
amount of arithmetic, there will be an arithmetical sentence such that
neither it nor its negation is provable. In Carnap's terminology, this
seems to yield non-determinate sentences, which is a problem for him
if we are convinced that nonetheless some of these sentences are true;
and indeed the key type of sentence used to prove the incompleteness
result—‘Gödel sentences’—are in fact true
in the standard model of arithmetic if these sentences are constructed
in an appropriate way (there are different ways of doing this) from a
theory itself true in that model.

Gödel himself wrote, but did not publish, a searching critique
of Carnap's position (Gödel, 1953–9/95). Gödel focuses
not on his first incompleteness theorem but on the corollary he drew in
his second theorem: that, under a certain natural characterization of
the property of consistency, a characterization which can be given
mathematically via his arithmetization of syntax, no formal theory of
the type Gödel considered could prove its own consistency.
He argued that Carnap, in order to make good his positivistic
thesis that mathematical theorems are devoid of content, needed to give
a consistency proof for mathematical calculi in order to show that they
do not have empirical content, an abundance thereof indeed, by dint of
entailing all empirical sentences. Warren Goldfarb notes, however,
(1995, p. 328) that this point fails to appreciate the deep holism of
Carnap's 1937 position where the distinction between analytic and
synthetic is relative to the system in question, the ‘linguistic
framework’. (This deep holism, of course, has the
counter-intuitive consequence that there is no framework-transcendent
distinction between mathematics and empirical sciences.)

Carnap, in fact, understood the import of Gödel's theorems
(Tennant, 2008); he knew of the results directly from Gödel, who
indeed read drafts of the Logical Syntax. Despite this he
displayed what looks like remarkable insouciance with respect to its
implications for his position. He acknowledged the need, in
demonstrating consistency, to move to a stronger language (§60c),
and freely helped himself to mathematical techniques which could in no
sense be classed as finitary (in §14 he used, for example, rules
with infinitely many premisses, notably one which was later to be
called the ω-rule). In this way he can deny, for arithmetic at
least, that there are any non-determinate sentences since every
true arithmetic sentence is provable using the ω-rule (relative to a
fairly weak finitary logic, considerably weaker than classical
logic).

Carnap's relaxed attitude stems from his abandonment of the
search for epistemological foundations. If one hopes to secure our
knowledge of mathematics by appeal to a formalist interpretation then
the search for a consistency proof of the type Hilbert sought makes
sense and one will look for a vindication of mathematics as a whole
from within a limited fragment with respect to which our knowledge
seems hard to impugn. But Carnap, perhaps as a result of
Gödel's deep theorems, seems to have abandoned that goal
and thought the Principle of Tolerance absolved him of any such
need. One can stipulate what one likes, including stronger axiom
systems from which one can prove the consistency of a weaker theory.
This does not give any firmer grounds for believing or accepting the
weaker theory, but one does not need such grounds anyway.

Few nowadays look for Cartesian certitude in mathematics, so
Carnap's position here may seem reasonable. It is not so clear,
however, that he has answered the problem of applicability. Even if a
conservative extension result can only be given in a more powerful
system, we need the result to be a contentful truth, not just a string
of symbols we can derive from some system, if we are to have
reassurance that a particular calculus we are about to use in designing
bridges or computers will be pragmatically useful. And if the Carnapian
grants that the result is a contentful truth, we can ask what,
according to this formalistic position, constitutes that truth. Carnap,
to be sure, was motivated by a horror of becoming embroiled in
metaphysical disputation. But if his standpoint is emptied not only of
epistemological ambition but is so deflationistic as to say little more
than that metamathematical techniques can be applied to formalisations
of mathematical and scientific theories then it is also emptied of all
philosophical interest and ceases to make an intervention in the
debates in philosophy of mathematics.

Not that Carnap really is abandoning metaphysics: this erstwhile
opponent of metaphysics is really a brother metaphysician with a rival
theory of his own, as F. H. Bradley might have said. Thus the main
import of the later ‘Empiricism, Semantics, and Ontology’
is the dismissal of ontological worries as pseudo-problems by
dint of the (highly contentious) distinction between
‘internal’ questions, to be settled by the rules of the
framework—of mathematics, ordinary ‘thing’ talk or
whatever—and ‘external’ questions such as
‘which framework to adopt?’. To these external questions
there correspond, Carnap claimed, no propositions with truth values; to
no such question, for example, is there a true answer of the form:
“Yes, infinitely many abstract numbers exist”. Rather, they
are to be answered by decisions, decisions to adopt or not
based on pragmatic criteria regarding the efficiency, fruitfulness and
utility of the framework with respect to the aims of the discourse in
question. But what aims could these be? Predicting the flow of
‘sense data’, taken for granted as forming the ultimate
furniture of the world? If so, we see that the vaunted ontological
neutrality is a sham and we have a radical form of empiricist
anti-realist metaphysics.

The vestige of formalism lies in this: Carnap takes
‘correctness’ to be determined by rules which govern a
system and not to consist, for example, in correspondence with a realm
of facts independent of the system of rules. And he thinks this
approach dissolves ontological worries and frees us from any obligation
to explain how finite, flesh-and-blood creatures like ourselves could
come to have detailed knowledge of this independent realm of facts,
this domain of configurations of abstract, non-temporal, non-causal objects
and properties, a realm exposed by Carnap as a metaphysical illusion.
One strong disanalogy with formalism is that Carnap takes this line
with all areas of discourse, not just mathematics.

W. V. Quine famously rejected the positivist's doctrine of truth in
virtue of meaning and the quasi-logicist conception of mathematics as a
body of analytic truths (and, partly as a result, also rejected
his mentor Carnap's internal/external distinction). Quine, in
conjunction with Nelson Goodman, produced instead what amounts to a
formalist manifesto. His formalist phase does not seem to have lasted
very long: he later settled on a form of mathematical platonism,
downplaying, if not largely ignoring, his relatively youthful
flirtation with ‘nominalism’. But while it lasted he and
Goodman greatly advanced the discussion of formalism
by tackling head-on the questions which other formalists shirked or
ignored.

The gains which seem to have accrued to natural science from the use
of mathematical formulas do not imply that these formulas are true
statements. No one, not even the hardiest pragmatist, is likely to
regard the beads of an abacus as true; and our position is that the
formulas of platonistic mathematics are, like the beads of an
abacus, convenient computational aids which need involve no question
of truth. (p. 122).

They regard

the sentences of mathematics merely as strings of marks without
meaning

so that

such intelligibility as mathematics possesses derives from the
syntactical or metamathematical rules governing those marks. (p.
111)

Commendably, Goodman and Quine do not shy away from the metatheory
problem, the difficulty that syntax and metamathematics itself seems as
ontologically rich and committed to abstract objects as arithmetic. On
the contrary, they face it squarely and attempt to make do entirely
with an ontology of concrete objects, finitely many such objects in
fact. (However they do assume fairly powerful mereological principles,
essentially universal composition: they assume that any fusion of
objects, however scattered or diffuse, is also an object in good
standing.)

With much ingenuity they try to develop a syntax which “will
treat mathematical expressions as concrete objects”
(ibid.)—as actual strings of physical marks—and
give concrete surrogates for notions such as ‘formula’,
‘axiom’ and ‘proof’ as platonistically defined.
However they do not address the issue of the application of
mathematics, construed in this concrete, formalistic fashion.

In addition to the applicability problem, there are two further crucial
problems for formalism as developed by Goodman and Quine. Firstly it is
not clear that they are entitled to the general claims they make about
syntax, construed as a theory about certain concrete marks and fusions
of marks. Thus when arguing that their definition of formula in terms
of ‘quasi-formula’ gives us the results we want, they
say:

By requiring also the next more complex alternative denials in
x to be alternative denials of quasi-formulas, the
definition guarantees these also will be formulas in the intuitively
intended sense; and so on, to x itself. (p. 116)

(‘Alternative denial’ is the Sheffer stroke operation
P|Q which is true if and only if one component is
false.) The problem lies with the ‘and so on’. Goodman and
Quine are trying to work their way up through an arbitrary formula
showing that their definition will ensure that each larger component
is a formula. It is not clear how we can have a guarantee of this for
arbitrary x, without something like induction over formula
complexity; but this is not available as formulas are not generated in
the usual inductive set-theoretic fashion. Similar remarks apply to
the demonstration that proofs, as defined nominalistically on p. 120,
have the internal order of precedence among immediate sub-premisses
and conclusions that we intuitively expect. The demonstration proceeds
with a generalisation over all numbers k which number the
axioms in a concrete proof and then performs a sequence of selections
on them. This seems to presuppose the truth of generalisations over
all numbers and indeed countable choice, resources unavailable to a
strict nominalist.

Secondly, what can Goodman and Quine say about a sentence such
as

222222+1 is
prime ?

(That is—with ‘2^n’ representing ‘2
to the power n’—‘[2^(2^(2^(2^(2^2))))]+1 is
prime’; cf. Tennant, 1997 p. 152.) They cannot deny the sentence
exists, for there is the token before our very eyes. But there are
strong grounds for thinking that no concrete proof or disproof will
exist, for the only methods available may use up more time, space and
material than any human could have at her disposal, perhaps than
actually exists. There are countless sentences with this property:
concrete tokens of them exist but no concrete proof or refutation
actually exists, none that a human could manipulate as a meaningful
utterance anyway. (Cf. George Boolos' ‘A Curious
Inference’, 1987.) Formalists of Goodman and Quine's persuasion
seem forced to the conclusion that sentences like the above, sentences
which are decidable in the usual formal sense, are neither true nor
false, since neither (concretely) provable nor refutable. To embrace
this view, however, would be to butcher mathematics as currently
practiced; such a consequence should rather be viewed as
a reductio ad absurdum of their position.

The most substantive attempt at a non-Hilbertian formalist philosophy
of mathematics is Haskell Curry's book Outline of a Formalist
Philosophy of Mathematics (Curry, 1951). Curry is no game
formalist, his position is closer to term formalism, of the two views
we started out from. Curry's philosophy of mathematics, however,
is, or attempts to be, a highly anti-metaphysical one, at least to the
extent that he thinks he can remain neutral on the issue of the
ontological commitments of mathematics.

Mathematics can be conceived as a science in such a way as to be
independent of any except the most rudimentary philosophical
hypotheses. (p. 3)

Hence he is not motivated by an anti-platonist horror of abstract
objects. His neutrality, indeed, is somewhat compromised by
the fact that Curry is perfectly happy to commit to an infinite
ontology of presumably abstract expression types. Officially he evinces
disinterest in what the primitives—he misleadingly calls
them ‘tokens’—of his formal systems are

We can take for those tokens any objects we please, and similarly we
can take for operators any ways of combining these objects which
have the requisite formal properties. (p. 28)

But since for many systems there are infinitely many primitive
‘tokens’, they cannot all be identified with concrete marks
which mathematicians have actually produced.

Like the term formalist, Curry takes mathematics, properly
reconstructed after philosophical reflection, to have an essentially
syntactic subject matter, namely formal systems. Unlike Frege's
adversaries, though, Curry, writing after the development of the
discipline of metamathematics, is able to give a far more rigorous
(albeit in his case somewhat eccentric) account of what a formal system
is.

No restrictions are placed on what form the axioms, rules and
therefore theorems of a formal system are to be. Truth for elementary
propositions of a formal system consists simply in their provability
in the system. One of his formal systems (Example 7, p. 23) has only
one predicate “a unary predicate expressed by Gödel by the
words ‘ist beweisbar’” (p. 23), i.e., a provability
predicate. The elementary truths of this system can be interpreted as
claims about provability in the underlying system. Any formal system
of the usual sort could be ‘reduced’ into one in which
there is only one provability predicate and truth (= provability) in
the reducing system of the elementary proposition ⊢ ⟨φ⟩
holds only when φ is provable in the reduced system
(pp. 34–35). Curry allows that one can form compounds from elementary
propositions by means of the usual logical operators in order to
express complex propositions in the language of proof theory (Chapter
IX).

The upshot is that mathematics in general becomes metamathematics, a
contentful theory—Curry's sentences express propositions with
truth values—setting out the truths about what is provable from
what in underlying formal systems whose interpretation, or rather
interpretations, are not taken to be mathematically important. This
standpoint, however, threatens to collapse into structuralism, into
viewing mathematical utterances as schemata implicitly generalising
over a range of (in general) abstract structures which satisfy
the schemata. As to the problem of the metatheory, Curry does not seek
to answer this; there is no real attempt to avoid commitment to a rich
ontology of objects, except that, by considering only standard formal
systems, one could make do with a countable ontology which can play the
role of linguistic expressions. Only, however, at the cost of
grossly distorting mathematical practice. Set theorists, topologists,
analysts et al. entertain conjectures, and try to prove things
‘about’ sets, topological spaces, functions on the complex
numbers and so on. In their philosophical moments they may wonder just
what the concepts they are wrestling with are ‘about’, but
they do not in general entertain conjectures or try to prove things
‘about’ expression strings, except where it is of
instrumental value to them in proving things about sets, spaces, the
complex plane and so on (cf. Resnik pp. 70–71).

Later developments have been primarily in the
‘Hilbertian’ wing of the formalist movement. P. J. Cohen's
work on the generalised continuum hypothesis showed, in tandem with
Gödel's relative consistency proof, that it and countless related
set-theoretic propositions concerning the relations of the cardinality
of a set to that of its powerset are undecidable by current
axioms. With no obvious, non-ad hoc, ways to extend the axioms to
decide these questions, this led some mathematicians, such as Cohen
himself (Cohen, 1971) and Abraham Robinson, (Robinson, 1965, 1969) to
despair of a realistic interpretation of higher set theory. They
therefore treat branches of mathematics in which no plausible axiom
set will decide the key questions as ‘ideal’ parts of
mathematics, lacking the content to be found in other areas.

As to game formalism, although philosophers may accuse mathematicians
of a tendency to lapse into this discredited position, very few
philosophers advance views resembling the game formalists. However
Azzouni (2005) has defended a viewpoint in mathematics he describes as
formalist, though see also his (2009), and Gabbay (2010) defends a
formalist position whilst Weir's approach (1991, 1993, 2010) is
clearly in the game formalist camp.

The latter's position starts from a fairly common
‘post-Fregean’ or ‘neo-Fregean’ perspective on
language. Frege, at least early in his career, held that the truth
value of a sentence is determined by two factors,
the sinn—the sense, literal meaning or informational
content, of the sentence—and the way the world is. Indexicality
and wider context relativity of sentences he initially supposed could
be met by assuming speakers who uttered and grasped such sentences
took them as elliptical for more complete utterances whose sense
fixed, in combination with the world, a unique truth value.

Later work (including Frege's own) revealed the inadequacy of
this picture, revealed that some indexicality, for instance, is
‘essential’ in John Perry's phrase, to the thought
expressed. I can utter “it's hot now” truly without
knowing where, when or even who (if sufficiently disoriented or out my
head) I am. Those who are not utterly sceptical, as radical
contextualists are, of systematic theories of meaning, will amend
Frege's view to a tripartite one. A sentence's truth value,
in a particular context, is determined by its informational content, by
contextual circumstances related to, ‘fitted to’ the
utterance by aspects of the speakers’ practice, and finally
by the mind and language-independent world. The contextual
circumstances need not figure in the sense or informational content of
the utterance; thus a specification of them may include dates and
locations, though this is not part of the sense of “it's
hot now”.

This picture in turn suggests the idea that the contextual
circumstances which ‘make true’, in conjunction with
independent reality, an utterance, may make it true in a non-realist
fashion. For instance, in cognitivist forms of expressivism or
projectivism, internal sentiments may make true “that haggis is
tasty”, though reference or conceptualisation of those sentiments
is no part of the literal meaning of ‘tasty’. Likewise in
Weir's version of game formalism, what makes true (or
false) “sin2θ + cos2θ = 1” in a
particular system is the existence of a concrete proof or refutation,
though the statement that such a concrete proof exists is no part of
the literal meaning or sense of the claim.

The advantage of this type of formalism is that it not only affirms the
meaningfulness, the possession of sense, of mathematical utterances; in
sharp contrast with traditional game formalism, it holds that such
utterances have truth values, where proofs or refutations exist. The
problem of the metatheory is met if one can give, after the fashion of
Goodman and Quine, a non-mathematical account of concrete proof. Of
course severe problems remain. The problem of applicability has to be
met, by providing conservative extension proofs for example. And of
course the theory is threatened not only by Gödel-type
incompleteness and the undecidable—but intuitively
truth-valued—sentences it throws up; even more devastating
indeterminacy looms in the form of ‘concretely undecidable’
sentences such as the primality claim above. The formalist who espouses
the meaningfulness and indeed truth of standard mathematical theories,
including proof theory for example, has more resources to meet such
claims than classic game formalism; the question is whether these are
enough to salvage a position most still think hopeless.